Can a purely policy convolution neural network based game learn to play better than its opponents?
(I'd leave this in the comments but sadly I can't.)

Here's a great paper (leveraging formal game theory) in which "players" make strategic choices based on each event ("move") and an algorithm calculates the best strategy for all players based on each player's abilities throughout the multi-stage "game". So instead of learning from winning and predicting what the next perfect move the player thinks (based off of learning) will lead to a "win", the player calculates an optimal strategy and moves to win the game itself over multiple moves. 

https://users.ece.cmu.edu/~youzhib/paper/bao2017csf.pdf

Note: This paper deals with cyber war games so is heavily tilted towards that domain, but if you take away the game theoretics and concept of strategy over a multi-stage game like ti-tac-toe, you should be able to improve your overall outcomes. That being said, how the model would accommodate a random "player" with random moves would be interesting to see.